Digital Image Processing: Image Enhancement Frequency domain methods - Duong Anh Duc

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Digital Image Processing: Image Enhancement Frequency domain methods - Duong Anh Duc provides about Image Enhancement - frequency domain methods; 1-d Fourier transform of a sequence; 2-d Fourier transform of a digital image; Fourier Transform of “Rice” Image; Importance of Phase Information in Images;...

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Nội dung Text: Digital Image Processing: Image Enhancement Frequency domain methods - Duong Anh Duc

Digital Image Processing Image Enhancement Frequency domain methods 21/11/15 Duong Anh Duc - Digital Image Processing 1
Image Enhancement: Frequency domain methods  The concept of filtering is easier to visualize in the frequency domain. Therefore, enhancement of image f(m,n) can be done in the frequency domain, based on its DFT F(u,v) .  This is particularly useful, if the spatial extent of the point-spread sequence h(m,n) is large. In this case, the convolution PSS Enhanced Image g(m,n) = h(m,n)*f(m,n) Given Image may be computationally unattractive. 21/11/15 Duong Anh Duc - Digital Image Processing 2
Frequency domain methods  We can therefore directly design a transfer function H(u,v) and implement the enhancement in the frequency domain as follows: Transfer Function Enhanced Image G(u,v) = H(u,v)*F(u,v) Given Image 21/11/15 Duong Anh Duc - Digital Image Processing 3
1-d Fourier transform of a sequence  Given a 1-d sequence s[k], k={…,-1,0,1,2,…,}  Fourier transform  Fourier transform is periodic with 2  Inverse Fourier transform 21/11/15 Duong Anh Duc - Digital Image Processing 4
1-d Fourier transform of a sequence  How is the Fourier transform of a sequence s[k] related to the Fourier transform of the continuous signal  Continuous-time Fourier transform 21/11/15 Duong Anh Duc - Digital Image Processing 5
2-d Fourier transform of a digital image  Given a 2-d matrix of image samples s[m,n], m,n Z2  Fourier transform  Fourier transform is 2 -periodic both in x and y  Inverse Fourier transform 21/11/15 Duong Anh Duc - Digital Image Processing 6
2-d Fourier transform of a digital image  How is the Fourier transform of a sequence s[m,n] related to the Fourier transform of the continuous signal  Continuous-space 2D Fourier transform 21/11/15 Duong Anh Duc - Digital Image Processing 7
Fourier Transform Example f(x,y) |F(u,v)| displayed as image 21/11/15 Duong Anh Duc - Digital Image Processing 8
Fourier Transform Example |F(u,v)| displayed in 3-D 21/11/15 Duong Anh Duc - Digital Image Processing 9
Fourier Transform ExampleImage Image Magnitude Spectrum 21/11/15 Duong Anh Duc - Digital Image Processing 10
Fourier Transform ExampleImage Image Magnitude Spectrum 21/11/15 Duong Anh Duc - Digital Image Processing 11
Fourier Transform ExampleImage Image Magnitude Spectrum 21/11/15 Duong Anh Duc - Digital Image Processing 12
Fourier Transform ExampleImage  As the size of the box increases in spatial domain, the corresponding “size” in the frequency domain decreases. 21/11/15 Duong Anh Duc - Digital Image Processing 13
Fourier Transform of “Rice” Image f(x,y) |F(u,v)| 21/11/15 Duong Anh Duc - Digital Image Processing 14
Fourier Transform of “Rice” Image F(u,v) 21/11/15 Duong Anh Duc - Digital Image Processing 15
Fourier Transform of “Camera Man” Image g(x,y) |G(u,v)| 21/11/15 Duong Anh Duc - Digital Image Processing 16
Fourier Transform of “Camera Man” Image G(u,v) 21/11/15 Duong Anh Duc - Digital Image Processing 17
Importance of Phase Information in Images  Image formed from magnitude spectrum of Rice and phase spectrum of Camera man 21/11/15 Duong Anh Duc - Digital Image Processing 18
Importance of Phase Information in Images  Image formed from magnitude spectrum of Camera man and phase spectrum of Rice 21/11/15 Duong Anh Duc - Digital Image Processing 19
1-D Discrete Fourier Transform (DFT)  For discrete images of finite extent, the analogous Fourier transform is the DFT.  We will first study this for the 1-D case, which is easier to visualize.  Suppose { f(0), f(1), …, f(N-1)} is a sequence/ vector/1-D image of length N. Its N-point DFT is defined as  Inverse DFT (note the normalization): 21/11/15 Duong Anh Duc - Digital Image Processing 20